In a similar vein, the phase-space formalism, initially developed by Groenewald [1]} and independently by Moyal [2]}, building on ideas from Wigner [3]}, among others [4]} involves converting the master equation formalism into a description of the evolution of quasiprobability distribution on phase space. This has been used very successfully to describe a wide variety of applications, such as the effects of decoherence in matter-wave interferometry [5]}, modelling the behaviour of quantum walks as a basis for novel quantum algorithms [6]}, modelling fundamental tests of quantum mechanics in cavity QED [7]}, and even furnishing descriptions of coherent dynamics in the early universe [8]}. The flexibility of phase-space approaches is hindered only, as in the case of the density matrix formalism, by the growth of the computational complexity of numerical propagation in higher dimensional settings. This is because the Wigner-Weyl transformation between the density matrix description and the Wigner function description is invertible, and so both necessarily contain the same number of degrees of freedom.
